hmwk5 - = R ◦ exp ◦ S ◦ T . (A diagram would help to...

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Math 220A Complex Analysis Solutions to Homework #5 Prof: Lei Ni TA: Kevin McGown Conway, Page 54, Problem 7. If T ( z ) = az + b cz + d , then ﬁnd z 2 ,z 3 ,z 4 (in terms of a , b , c , d ) such that T ( z ) = ( z,z 2 ,z 3 ,z 4 ). Proof. We have T - 1 ( z ) = dz - b - cz + a , and we compute T - 1 (1) = d - b a - c T - 1 (0) = - b a T - 1 ( ) = - d c . Therefore T ( z ) = ( z,z 2 ,z 3 ,z 4 ) with z 2 = d - b a - c , z 3 = - b a , z 4 = - d c . ± 1

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Conway, Page 55, Problem 14. Suppose one circle is contained inside an- other and that they are tangent at the point a . Let G be the region between the two circles and map G conformally onto the open unit disk. Rough Proof Sketch. First apply the M¨ obius map T ( z ) = ( z - a ) - 1 . This sends G to the region between two parallel lines. Using a rotation followed by a translation, we can send this region to the region between any two parallel lines we like. That is, we may choose S ( z ) = cz + d with | c | = 1 so that S ( T ( G )) = { x + iy | 0 < y < π/ 2 } . Hence exp( S ( T ( G ))) = { x + iy | x > 0 } . Finally, the M¨ obius map R ( z ) = z - 1 z + 1 sends the right half-plane onto the unit disk D . Therefore f ( G ) = D when we set f
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Unformatted text preview: = R ◦ exp ◦ S ◦ T . (A diagram would help to illustrate what f is doing.) It is easy to check that f is conformal, being a composition of conformal maps. More explicitly, we can write f ( z ) = exp ± c z-a + d ²-1 exp ± c z-a + d ² + 1 . The constants c and d will depend on the initial location of the two circles in the complex plane. ± 2 Conway, Page 67, Problem 6. Show that if γ : [ a,b ] → C is a Lipschitz function then γ is of bounded variation. Proof. Suppose γ has Lipschitz constant M . Let P = { a = t < t 1 < ··· < t m = b } be any partition of [ a,b ]. Then we have v ( γ ; P ) = m X k =1 | γ ( t k )-γ ( t k-1 ) | ≤ m X k =1 M ( t k-t k-1 ) = M ( b-a ) . Since P was arbitrary, we conclude that γ is of bounded variation. ± 3...
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This note was uploaded on 10/19/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.

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hmwk5 - = R ◦ exp ◦ S ◦ T . (A diagram would help to...

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